基于坐标转换的面对面定向误差最小二乘评定

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Sep．2013机 床 与 液 压Hydromechatronics Engineering V01．41 No．18DOI：10．3969／j．issn．1001—3881．2013．18．014Calculation of Plane to Plane Orientation Error by LeastSquare Method Based on Coordinate TransformationW ANG Xu ，HUANG Meifa，CHEN Leilei，YANG ZhengSchool ofMechanical&Electrical Engineering，Guilin University ofElectronic Technology，Guilin 541004，China1．IntroductionAbstract：Currently Least Square Method(LSM)is commonly applied in the error evaluation，butit is adaptive to the low accuracy and the obtained value is not the minimum．1fl order to accurate.

1y evaluate orientation error，this paper presents a method by using the obtained optimum associ—aled datum plane．This method rotates the initiaI associated datum plane as far as possible to ap.

proach the minimum zone by least square method based on coordinate transformation．The eva1.

uation model of plane to plane orientation error is established uniformly by adjusting the ideal rightangle．Finaly，we adopt the evaluation example of plane measurement to plane paralelism errorto validate the proposed method。The experimental results confirm that this method is correct.

Key words：orientation error，optimum associated datum plane，coordinate transformation，leastsquare methodThe orientation error is one of the importantspecifcations in the geometry measurement of theprecision pans and has a significant influence on thequality and performance of product．In practical ge—ometry measurement，it is very impo~ant to accurate—ly and quickly obtain the evaluated results.

Curently，there are two methods to evaluate theorientation eror， they are the traditional measure.

ment method and least square method．The tradition—al measurement method has low measurement preci.

sion and is not capable to obtain the accurate eror.

Least Square Method is simple and quick but thismethod is only for the approximate evaluation and cannot be applied for minimum zone evaluation． In aword，the existing measurement methods do not meetthe requirement of the modern advanced manufactureReceived：2012—12—26Guangxi Key Laboratory of Manufacturing Systems and Ad—vanced Manufacturing Technology(1 1—031—12_004)WANG Xu．E·mail：waitwang123### 163．comindustry 『1—2].

The transformation of COOrdinate system is aprocess in which the rotation and translation of thecoordinate system are used to obtain the evaluatedvalues．This paper presents a least square methodbased on the transform ation of coordinate system to e—valuate orientation e11"or.

2．Establishment of the optimum associateddatum planeThe general process of this method includes twosteps．First，we apply Least Square Method(LSM)toassociate the initial datum plane．Second，we makethe flatness error of the initial associated datum planeas far as possible to approach the minimum zone byusing the coresponding transformation[3].

Assume that the measured point of arbitrary spa—tial orientation in the actual plane are P ( ，Y ，z )(i=1，2，?，n)．The equation ofthe initial associ—ated datum plane Mo could be expressed in Equation(1)：Z +my+nz+h=0 (1)The distance d from pointP ( ，Y ， )(i=1，WANG Xu，et al ：Calculation of Plane to Plane Orientation Error bv LeastSquare Method Based on Coordinate Transformation2，?，／'t)totion(2)：plane M0 could be expressed as Equa一= h+ lxi +m鬲yi+nz (2)Where，d is the function of l，m，n． The norm alvector of plane M0 is(f，m，n)．According to thedefinition of least squares associated plane
． the sumof squares of distance from point P ( ，Y ，z )(i=1，2，?，n)to plane M0 is the minimum，we takethe sum of squares function as the optimal targetfunction：f(1，m，n)=∑d (3)The values of l，，n and n could be obtained bvthe equation of the least square plane． The flatnesserror F is the difference between the maximum dis—tance and minimum distance from the measuredpoints to the ideal datum plane：n nF = MA
，X(d )一MI (d ) (4)W e take distance between measured points andplane as positive if these points are above the leastsquares associated plane，and the distance as nega.

tive if these points are below the least squares associ.

ated plane.

y (a)P，(b)Fig．1 Relationship diagram of point Pnplane M0 and M1As shown in Fig．1(a)，the initial associateddatum plane Mo is projected respectively in the direc—tions which are perpendicular to the direction of toler.

ance zone and the direction of ligature of the highestcontact point and the lowest contact point of tolerancezone(as shown in Fig．1(b))，the equation of theinitial associated datum plane M0 is already obtained.

The normal vector T of plane M0 is(ao，b0，co)andthe flatness error of the plane M0 is F
． Through cal—culation and analysis，we rotate plane M0(rotate thenormal vector of plane M0)to decrease the value of F.

consciously and slightlyThe optimum associateddatum plane M1 is acquired when the plane M0 isslightly rotated.

Th e corresponding point of point Pf is P ． Thedistance from point P ( ，Y ， )(i=1，2，?，n)to plane M0 is F ．The maximum and minimum valueofFk is Fi and ，respectively．Fk=Fi一 ： P Bo— P A0．The point B1 can be obtained from the dis.

tance of占( is a very smal value)based on thepoint A0 along A0P ，and the point Al can be obtainedfrom the distance of based on the point Bo alongB0P ．Both point A1 and point B1 belong to the opti—mum associated datum Plane M1． The distance be.

tween point P and plane M1 is F ． The maximumand minimum value of F t is F t and F ，respective.

1y．Fk =F 一F'j= P Al—PfB1．Fk ，
thevalue of eror F could be decreased[3].

3．Establishment of orientation error evalu.

ation modelW hen tolerance t of given parallelism，perpen-dicularity or angularity is 0，the ideal angle is 0。
，90。or a given certain angle，respectively．When theactual angle is deviated from the ideal angle，the par.

allelism，perpendicularity or angularity appears，re.

spectively．Size of the deviation is the eror of eachprojected tolerance[4].

As shown in Fig．2，the model of plane to planeorientation eror evaluation is established uniformlyby changing the ideal right angle Q．The tolerancezone direction of plane to plane orientation toleranceis unique．After the position of the datum feature isdetermined，two paralel planes(tolerance zone)oftolerating measured plane is also determined[4].

Fig．2 Plane to plane orientation error evaluation modelAssociated plane of actual measured plane 1 isplane M． The equation of the associated plane M72 Hydromechatronics Engineeringcould be expressed as Equation(5).

Ax+By+Cz+D =0 (5)According to the optimum associated datumplane method，we evaluate the normal vector T ofplane M which is(A，B，C)．The normal vectorof datum plane 3 is(1，m，／,)．The intersection lineof plane M and datum plane 3 is line L．the directionvector T of line L could be expressed as Equation(6).

T=(A B C)×(1 m n) (6)QDatum plane 3 is revolved by ideal correct anglearound line to obtain auxiliary plane 2.

As shown in Fig．3，coordinate system rotationmatrix could be derived．Line L will be rotated to theposition in which it is parallel to the Z axis.

Fig．3 Coordinate system rotating figureAssume the direction vector T of line L be T=(u b c)．The coordinate of point 01 in line L is( ，Y ，z )．The coordinate of point Ol is acquiredby calculation model of coordinate rotation．The coor—dinate of point Ol is( ，Y ，z )．The calculationmodel of coordinate rotation could be expressed as E—quation(7)[5]：盼蜘 ；sjn 一 u b，eos咖=‘ 0 一sinT]1 0 l(7)0 cosv J。’． 、／0 +b c吼 一 ，∞ 丽 The coordinate rotationEquation(8)and(9).

r cosq~M=l—si呻 【0厂M1=f L_
matrix is expressed as01 r cosT 0 一siny]0Il 0 1 0 l(8)1 J【siry 0 c。s Jcos 1 si“ 1 0 ]0一
0
叩 j 1 J(9)As shown in Fig．2，the line L is rotated to theposition in which it is parallel to Z axis．LiBe isacquired by calculation model of coordinate rotation.

The direction vector of line L is T =TM． The nor—mal vector of datum plane 3 is T3 ， = TsM．Da—tum plane 3 is revolved by ideal corect angle Q a—round line L to obtain auxiliary plane 2．The norm alvector of auxiliary plane 2 could be expressed asEquation(10).

r l m n2=T'3M =I【r cosq~ 【一(10)Where， 1 is equal to Q．The eror value of the actu‘al measured plane l to datum plane 3 is transformedinto the distance d between each measured point P( ，Y ， )(i=1，2，·一，n)and auxiliary plane2．The difference of maximum distance and minimumdistance is the eror.

When the ideal fight angle Q is 0。．we can 0b—tain parallelism eror according to the above method.

When the ideal right angle Q is 90。，we can obtainperpendicularity eror based on the above mentionedmethod.

4．Verification of an exampleParallelism error evaluation of a box is taken asan example to verify the application of the method asshown in Fig．4．W e use Coordinate Measuring Ma—chine(CMM)to acquire discrete point coordinates ofthe datum plane and the measured plane．The paral—lelism data of the measured parts is shown in Tab．1.

In order to evaluate the orientation eror，the su-periority of the diferent algorithm is mainly embodiedin the optimization of datum feature．The values of o—rientation error obtained by different evaluation meth—ods are not comparable.

Fig．4 A box structure呻o ．一∞叫 一唧0_氢= 0 ∞
一 C0 1 0 0
.

0 .

ⅡC S O S C W ANG Xu，et al：Calculation of Plane to Plane Orientation Error by LeastSquare Method Based on Coordinate Transformation 73Firstly，we apply traditional measurement meth—od to evaluate plane to plane parallelism error by u-sing dial indicator．According to the measuring meth-od，the datum plane error is 0．403 mm，and the par—allelism eror is 0．037 mm[6].

Secondly，we utilize CMM(Hexagon，the modelis global 07 ·10 ·07)to evaluate plane to planeparalelism error． The datum plane error is 0．236mm， and the plane to plane parallelism error is0．017 mm[7].

Finally．according to the mathematical modelwhich is proposed in this paper，we can calculate theparallelism error by using the above method．The da.

turn plane error is 0．221 mm ，and the plane to planeparallelism error is 0．016 mm．The results of thesethree evaluation methods are compared in Tab． 2.

The comparison results show that the results obtainedby the method in this paper are better than those ofCMM and traditional measurement method，and theproposed method in this paper has strongest potentialapplication value.

Tab．1 Paralelism data of the measured Darts mmTab．2 Calculation results comparisonEvaluation metIloddatum plane parallelism5．ConclusionsIn this paper，we adopt least square method toevaluate the plane to plane orientation error based oncoordinate transform ation．Through the comparisonsof results．it shows that the present model is easy toprogram，and is capable to solve the error of parallel—ism，perpendicularity and angularity．Therefore，thismethod could be widely used in the real applications.

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重庆理工大学 a．机械检测技术与装备教育部工程研究中心，时栅传感及先进检测技术重庆市重点实验室；b．机械工程学院，重庆 400054摘要：利用时空转换思想，以时间基准测量空间位移量，借鉴国际上先进的位移测量技术手段，设计直线式时栅传感器 A／D转换 电路。采用 AD7298 BCPZ型 12位 A／D转换芯片与 STM32F407VGT6型ARM处理器相结合，利用嵌入式操作系统Linux软件编程，使得系统具有更好的可靠性与实时性。实验结果表明：采用所设计的 A／D转换电路，最小分辨时间为 2．44 ns，能更好地实现传感器的高速、高分辨率采样 ，实现了直线式时栅传感器的误差修正与补偿。通过 ，A／D转换 电路 的设计，为高精度直线式时栅位移传感器的研制提供 了技术支持。

关键词：A／D转换；ARM；时栅传感器；误差修正与补偿中图分类号：TP212(Continued on 73 page)基于坐标转换的面对面定向误差最小二乘评定王 续 ，黄美发，陈磊磊，杨 征桂林电子科技大学 机电工程学院，广西 桂林 541004摘要：为了更加精确地评定定向误差，提 出了一种基于坐标转换的最小二乘获得最佳基准平面的评定方法。该方法通过转换初始基准平面使其尽可能接近最小局域。通过改变理想正确角度统一建立了面对面定向误差的评定模型。最后，以评定面对面平行度误差为例进行验证。结果验证 了该方法的正确性和 可行性 。

关键词：定向误差评定；最佳拟合基准平面；坐标 系转换；最小二乘法中图分类号：TH11